Inverse Trigonometric Functions Important Questions.

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1. Q1Easy

   What is the principal value range of the function y = sin⁻¹(x)?

   • A.[-π, π]
   • B.[0, π]
   • C.[-π/2, π/2]
   • D.[0, 2π]
   Answer: [-π/2, π/2]

   Explanation: The principal value branch of sin⁻¹(x) is defined to ensure a unique output, which is the interval [-π/2, π/2].

2. Q2Easy

   Which of the following is the range of the function y = cos⁻¹(x)?

   • A.[-π/2, π/2]
   • B.[-π, π]
   • C.[0, π]
   • D.[-π, 0]
   Answer: [0, π]

   Explanation: The principal value branch of cos⁻¹(x) is defined to be the interval [0, π] to ensure uniqueness.

3. Q3Easy

   What is the domain of the function y = tan⁻¹(x)?

   • A.[-1, 1]
   • B.[0, π]
   • C.All real numbers (R)
   • D.[-π/2, π/2]
   Answer: All real numbers (R)

   Explanation: The function tan⁻¹(x) is defined for all real numbers, and its principal value range is (-π/2, π/2).

4. Q4Easy

   Which of the following is NOT a principal value branch of the cosecant inverse function?

   • A.[-π/2, 0) ∪ (0, π/2]
   • B.[0, π] - {π/2}
   • C.[-π/2, π/2] - {0}
   • D.[-π, π]
   Answer: [-π, π]

   Explanation: The principal value branch of cosec⁻¹(x) is [-π/2, 0) ∪ (0, π/2], excluding 0.

5. Q5Easy

   Which trigonometric function's inverse is reflected across the line y = x to obtain its graph?

   • A.y = cos(x)
   • B.y = tan(x)
   • C.y = sin(x)
   • D.y = sec(x)
   Answer: y = sin(x)

   Explanation: The graph of y = sin⁻¹(x) is obtained by reflecting the graph of y = sin(x) across the line y = x.

6. Q6Easy

   What is the principal value range of the secant inverse function?

   • A.[-π/2, π/2]
   • B.[-π, π]
   • C.[0, π] - {π/2}
   • D.[-π/2, π/2] - {0}
   Answer: [0, π] - {π/2}

   Explanation: The principal value branch of sec⁻¹(x) is [0, π] excluding π/2 to ensure uniqueness.

7. Q7Medium

   Which of the following intervals is the principal value branch for the inverse sine function (sin⁻¹ x)?

   • A.0 to π
   • B.-π to π
   • C.-π/2 to π/2
   • D.-π to π/2
   Answer: -π/2 to π/2

   Explanation: The principal value branch of sin⁻¹ x is defined to ensure a unique output, which lies within the interval [-π/2, π/2]. This restriction makes the function invertible and one-one.

8. Q8Medium

   What is the range of the inverse cosine function (cos⁻¹ x) when considering its principal value branch?

   • A.[-π/2, π/2]
   • B.[0, π/2]
   • C.[-π, π]
   • D.[0, π]
   Answer: [0, π]

   Explanation: The principal value branch of cos⁻¹ x is defined to ensure a unique output, which lies within the interval [0, π]. This interval ensures the function is invertible and bijective.

9. Q9Medium

   Which of the following is NOT an example of an inverse trigonometric function?

   • A.sin⁻¹ x
   • B.cos⁻¹ x
   • C.cosec x
   • D.tan⁻¹ x
   Answer: cosec x

   Explanation: Inverse trigonometric functions include sin⁻¹ x, cos⁻¹ x, tan⁻¹ x, cot⁻¹ x, sec⁻¹ x, and cosec⁻¹ x. The reciprocal of cosecant (cosec x) is not an inverse function but a reciprocal function.

10. Q10Medium

    What is the domain of the inverse tangent function (tan⁻¹ x)?

    • A.[-1, 1]
    • B.[-π/2, π/2]
    • C.All real numbers (R)
    • D.[0, π]
    Answer: All real numbers (R)

    Explanation: The inverse tangent function, tan⁻¹ x, is defined for all real numbers because the tangent function is bijective over its principal value branch (-π/2, π/2).

11. Q11Medium

    Which ancient mathematician introduced the sine function and made significant contributions to trigonometry?

    • A.Pythagoras
    • B.Euclid
    • C.Aryabhata
    • D.Archimedes
    Answer: Aryabhata

    Explanation: Aryabhata, an ancient Indian mathematician, made significant contributions to trigonometry, including the introduction of the sine function and its applications in astronomy.

12. Q12Medium

    Which property ensures that a function can be inverted?

    • A.Injectivity
    • B.Surjectivity
    • C.Bijectivity
    • D.Periodicity
    Answer: Bijectivity

    Explanation: A function must be bijective (both one-one and onto) to have an inverse. This ensures that each element in the codomain is mapped by exactly one element in the domain.

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